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Class Field Theory by Franz Lemmermeyer

By Franz Lemmermeyer

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For the first point, consider Dirichlet characters as homomorphisms χ : G −→ C× for G = Gal (K/Q); then Gχ = ker χ is a subgroup of G, and we say that χ is unramified at p if p is unramified in the fixed field of Gχ . Clearly every χ is unramified at the primes p m, and the principal character 1l is unramified everywhere since its fixed field is Q. The key to the proof is the observation that χ is ramified at p if and only of χ(p) = 0. Details will be added later. Notes The conjecture that every arithmetic progression a + mb for coprime integers a and m contains infinitely many primes goes back to Euler.

It is easy to see that the integers represented by Q are exactly the integers n for which there is an α ∈ b with nN b = N α. To each principal ideal (α) of this form there correspond w values of α; moreover we have already seen that these principal ideals are in bijection with the ideals a ∈ c−1 of norm m such that ab = (α) is principal. 1. Let Q = (A, B, C) be a quadratic form associated to the ideal a. Then a natural number n is represented by Q if and only if there is an integral ideal b ∈ [a]−1 with N b = n.

2. Assume that a1 , . . , at ∈ Z are independent modulo squares. Then for any choice c = (c1 , . . , ct ) of signs cj = ±1, the set Sc of primes p satisfying at a1 = c1 , . . , = ct p p has Dirichlet density δ(S) = 2−t . If we choose c1 = . . = ct = +1, then S = Spl(K/Q) for the multi√ √ quadratic number field K = Q( a1 , . . , at ). Since the independence modulo squares of the aj is equivalent to (K : Q) = 2t , we find that the set of 1 primes splitting completely in K/Q has Dirichlet density (K:Q) .

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